{
  "id": "1612.09229",
  "version": "v1",
  "published": "2016-12-29T18:07:58.000Z",
  "updated": "2016-12-29T18:07:58.000Z",
  "title": "An Erdös-Révész type law of the iterated logarithm for reflected Gaussian processes",
  "authors": [
    "K. Dębicki",
    "K. M. Kosiński"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1606.01502",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $B_H=\\{B_H(t):t\\in\\mathbb R\\}$ be the fractional Brownian motion with Hurst parameter $H\\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\\sup_{-\\infty<s\\le t}(B_H(t)-B_H(s)-(t-s))$, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\\mathbb P(\\mathscr E_f)=\\mathbb P(Q_{B_H}(t) > f(t)\\, \\text{ i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\\mathbb P(\\sup_{s\\in[0,f_p(t)]}Q_{B_H}(s)>f_p(t))/f_p(t)=O((t\\log^{1-p} t)^{-1})$, $\\mathbb P(\\mathscr E_{f_p})= 1_{\\{p\\ge 0\\}}$. Consequently, with $\\xi_p (t) = \\sup\\{s:0\\le s\\le t, Q_{B_H}(s)\\ge f_p(s)\\}$, for $p\\ge 0$, $\\lim_{t\\to\\infty}\\xi_p(t)=\\infty$ and $\\limsup_{t\\to\\infty}(\\xi_p(t)-t)=0$ a.s. Complementary, we prove an Erd\\\"os-R\\'ev\\'esz type law of the iterated logarithm lower bound on $\\xi_p(t)$, i.e., $\\liminf_{t\\to\\infty}(\\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$; $\\liminf_{t\\to\\infty}\\log(\\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\\in(0,1]$, where $h_p(t)=(1/z_p(t))p\\log\\log t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-29T18:07:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60G70",
      "60G22"
    ],
    "keywords": [
      "erdös-révész type law",
      "reflected gaussian processes",
      "stationary storage process",
      "fractional brownian motion",
      "iterated logarithm lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}